1. **State the problem:** Simplify the expression $$\frac{x + 1}{x(x - 1)} - \frac{1}{2(x - 1)}$$ and identify which of the given options it equals. 2. **Find a common denominator:** The denominators are $x(x - 1)$ and $2(x - 1)$. The least common denominator (LCD) is $$2x(x - 1)$$. 3. **Rewrite each fraction with the LCD:** $$\frac{x + 1}{x(x - 1)} = \frac{2(x + 1)}{2x(x - 1)}$$ $$\frac{1}{2(x - 1)} = \frac{x}{2x(x - 1)}$$ 4. **Subtract the fractions:** $$\frac{2(x + 1)}{2x(x - 1)} - \frac{x}{2x(x - 1)} = \frac{2(x + 1) - x}{2x(x - 1)}$$ 5. **Simplify the numerator:** $$2(x + 1) - x = 2x + 2 - x = x + 2$$ 6. **Final simplified expression:** $$\frac{x + 2}{2x(x - 1)}$$ 7. **Compare with options:** None of the options exactly match $$\frac{x + 2}{2x(x - 1)}$$, so check if any option can be equivalent by substitution or simplification. 8. **Check option (D):** $\frac{x + 1}{2x(x - 1)}$ is close but numerator differs. 9. **Conclusion:** The simplified form is $$\frac{x + 2}{2x(x - 1)}$$ which does not match any given option exactly. **Answer:** None of the options (A), (B), (C), or (D) equal the simplified expression. **Note:** If the problem expects the closest match, option (D) is similar but not equal.